Why is it some monks (Yuttadhammo specifically) walks barefoot in his alms rounds as well as while doing walking meditation?
Is there a purpose to it?
Buddha walked barefoot. There are some accounts that walking barefoot is good for your health but there are no Vinaya rules to address this issue. Ven Yuttadhaammo injured his foot be walking barefoot so you have to be very mindful if you walk barefoot.
While attending a retreat at a Burmese monastery (here in the States), I was instructed by the monk there to walk without my socks, because if provided better ability to get sensation in the feet.
Barefoot used to be normal -- and wearing shoes was disrespectful and/or a sign of feeling superior.
The Vinaya used to explicitly allow footwear in outlying areas -- and not in public.
Lay society has changed, so wearing shoes is now seen as polite, and the custom has adapted accordingly -- i.e. you might see monks with footwear now - similarly there are times and places where the climate isn't appropriate for bare feet.
But I don't think of bare feet as remarkable. I used to walk barefoot as a teen (not for those same reasons), obviously enough it's feasible.
In ancient India , most of sages used to walk barefoot like tirthankar jains , buddha , their best students. Walking barefoot was also among poor locals. Nowdays too , few forest tribals of North-East India , Chhattisgarh & M.P. live barefoot.
Among sages , barefoot living was an indication to :
Simple life (like common man)
Vairagya (no interest in pleasures)
Energy tranfer occurs through foot
To not kill small beings ,even unwillingly(slight chance) cause barefoot increases sensations.
Truth-Seeker has stopped too much travelling and gone forth for deep meditation within a small area.
Moreover ,as Meditator becomes strong in meditation , tendency towards barefoot increases, without any enforcement of sheel(Vinaya) .
It is also very practical because one needs not dry out shoes or socks, going barefoot or wearing sandals of some kind solves this. The staying dry comes into play a lot as it gets cold andor wet, it takes a lot of maintenance and winter shoes are weighty.
Afaik people can comfortably walk thus partially exposed to cold even in low temperatures like -20C but it requires some conditioning.
Basically It solves the problem of having more requisites than one can carry, in particular the 2-3 pairs of footwear & socks. One is saving ~3kg of weight there which is a lot considering that it's probably going to make up for ~20% of the total weight of requisites which can be comfortably carried.